Estimating neutrosophic finite median employing robust measures of the auxiliary variable

Our study explores neutrosophic statistics, an extension of classical and fuzzy statistics, to address the challenges of data uncertainty. By leveraging accurate measurements of an auxiliary variable, we can derive precise estimates for the unknown population median. The estimators introduced in this research are particularly useful for analysing unclear, vague data or within the neutrosophic realm. Unlike traditional methods that yield single-valued outcomes, our estimators produce ranges, suggesting where the population parameter is likely to be. We present the suggested generalised estimator's bias and mean square error within a first-order approximation framework. The practicality and efficiency of these proposed neutrosophic estimators are demonstrated through real-world data applications and the simulated data set.


The neutrosophic statistics
Neutrosophic statistics, a unique approach, are designed to handle datasets with a degree of ambiguity or partial information.This method allows for conflicting beliefs and accommodates a range of uncertain numbers that may represent some observations, including an exact measurement.In contrast, traditional statistics falter when faced with uncertainty.This is where the intriguing potential of neutrosophic statistics comes into play, offering a fresh perspective on data analysis.
In real-world problems, the population parameters are often unknown.In such cases, statistical inference methods may need to be more practical.Instead, acceptable estimates are used, resolving the issue of an unknown parameter value by estimating its values.This pragmatic approach reassures the statistician that the derived data are vague but still useful.Neutrosophic statistics, with their ability to calculate the best interval value with the minimum mean square error, offer a reliable solution to these problems.Previous study efforts provided a limited range of neutrosophic observations, including quantifiable neutrosophic data.Furthermore [32][33][34][35][36][37][38][39][40] discussed numerous approaches, such as interval-based approaches, Triangular or trapezoidal fuzzy numbers, and single-valued fuzzy numbers, exist to express the range of neutrosophic numbers along with Optimal trajectories in reproduction models of economic dynamics.He 41 proposed a fractal model for internal temperature response in porous concrete, advancing understanding in applied mathematics.Iskandarov and Komartsova 42,57 investigated integral perturbations' influence on boundedness in fourth-order linear differential equations.Khankishiyev 43 employed finite differences to solve loaded differential equations, while 44 , 56 explored dark energy solutions without a cosmological constant.Furthermore, 45 established conditions for complete monotonicity in the differential functions involving trigamma functions.
Let the neutrosophic range is , the neutrosophic variable T N indicates the neutrosophic samples selected from a population having imprecise, ambiguous and unclear measurements.Thus, for the neutrosophic data in the interval form, we use notation T N ∈ [a, b], where a and b are the lower and upper values of the neutrosophic data, respectively.
Figure 1 depicts the approach to applying the proposed estimation methods in neutrosophic statistics.This workflow developed a few neutrosophic estimators to estimate the finite population median in the presence of supplementary data, which are well suited for overcoming the sample indeterminacy problem.

The neutrosophic median estimators under simple random sampling
First, we present a few adapted neutrosophic median estimators using auxiliary information under simple random sampling to address uncertainty and neutrosophic data.

Adapted median estimators with auxiliary variable
(i) Motivated by 1 , we propose a neutrosophic traditional median estimator and its variance, along with the expression of variance is given by (ii) Motivated by 2 , we developed a novel neutrosophic traditional ratio estimator, along with the expressions of Bias and MSE are and The ratio estimator ( MRN ) performs better than M0N if ρ yxN > 0.5 (iii) Motivated by 46 , the neutrosophic exponential ratio-type estimator, along with the expressions of Bias and MSE are given by and The exponential ratio estimator ( MEN ) is more efficient than M0N and MRN if ρ yxN > 0.25 , respectively.
(iv) The adapted neutrosophic difference estimator along with the expression of variance is given by At the optimal value of d 0N , which is , the minimum MSE of MD 0N , is given by www.nature.com/scientificreports/(V) Adapted from 22 , difference-type estimators, along with the expressions of Bias and minimum mean square errors, are given by and where d iN (i = 1 − 8) are constants determined below by optimality considerations as The proposed generalized neutrosophic median estimator Traditional estimators, often hindered by their reliance on historical data, struggle with accuracy, particularly with outliers.This section introduces advanced neutrosophic estimators for accurately predicting a finite population's median.These estimators blend unique metrics like quartile deviation and interquartile range, enhancing data distribution analysis and outlier exclusion through robust scaling, employing decile means, the Hodges-Lehmann estimator, and tri-mean for reliable median estimation.The tri-mean proposed by 47 , the Hodges-Lehmann estimator proposed by 48 and the decile means proposed by 49 are the three robust metrics we used in this study.For further information about these robust measures, readers can see 50 and 51 for details.
are neutrosophic functions of the known robust and non-conventional measures related to the variable X N .Robust measures associated with X N are: The non-conventional measures (i.e., interquartile range, midrange, quartile average and quartile deviation) of the supplementary variable are as follows: (iv) Interquartile range: are the neutrosophic first, second and third quartiles, respectively and By putting different values of α i (for i = 3, 4) into ( 23), we get the following families of estimators as.When we use robust measures with linear combinations of the median, quartile deviation, midrange, interquartile range, and quartile average of the supplementary variable in (23), we get different series of estima- We can obtain several optimal estimators by placing suitable constants or known conventional parameters of the supplementary variable in place of ψ N and δ N into (23).Conventional parameters related to the supplemen- tary variable X N are variance, standard deviation, coefficient of variation, coefficient of skewness, coefficient of kurtosis, coefficient of correlation, and so forth. Vol The MSE of suggested estimator up to the first order of approximation as where ) , is given by ( 29)  Hence, robustness is evaluated in this case to compare the proposed neutrosophic generalized estimators with other neutrosophic estimators in (1), ( 3), ( 6), ( 9), ( 11), ( 12), (13), and ( 14) to find the more effective neutrosophic median estimator.Additionally, we use real-world datasets to determine the relative effectiveness of different estimators.

Real-life application
In terms of relative efficiency, we compare the suggested family of estimators' performance to that of other competitive estimators.We chose two real-world indeterminacy interval datasets for this purpose.
Regarding relative efficiency, we compare the suggested family of estimators' performance to that of other competitive estimators.For this purpose, we chose two real-world indeterminacy interval datasets.
The first one is the Daily stock price, which is used as a neutrosophic variable because, on each day, a stock's price fluctuates between an opening price (the price during which trade begins) and a closing price (the price at which trade stops for the day).The price constantly fluctuates between a high (the highest price of the day) and a low (the lowest price), which may or may not be similar to the opening or closing price.We estimate the high and low price intervals within which the stock price falls by utilizing the daily starting price as a supplementary variable that is not a neutrosophic variable since its value is set and known for each day.
Population II Source: 54 (1 st Feb 2022 to 29 th July 2022) from the link: https:// finan ce.yahoo.com/ quote/ SZKMY/ histo ry?p= SZKMY.Y N = Low & High prices; X N = Opening price, where Y N ∈ (Y L , Y U ) corresponds to the independent determinate variable X N ∈ (X L , X U ).
Figures 2 and 3 show the trend of real-world data sets using box plots, which aids in displaying the skewness of the data.The minimum score (Lower Fence), the lower quartile, the median, the upper quartile, and the maximum score summarise data using boxplots (Upper Fence).As shown in Fig. 2, our positively skewed data suggests that the median is closer to the lower quartile.The boxplot with points outside the whiskers shows a few outliers in the data.( 31) Vol.:(0123456789)In addition, the following formulae are used to get the percentage relative efficiency (PRE) 12).Table 3 presents the complete descriptions of each population mentioned below.Table 4 presents the complete descriptions of each population mentioned below.Tables 5, 6, 7, 8, 9 and 10 elaborate the PREs of all neutrosophic estimators relative to M0N .It is observed that the PREs of T i(d)N estimators change with the choices of α 3 and α 4 .It is further noted that the performance of T i(d)N is the best among all the estimators proposed here.
As indicated by the indeterminacy interval findings from (23) for the whole data set, the neutrosophic generalized estimator T i(d)N , is more efficient than the other suggested estimators studied.Also, the indeterminacy interval findings show that the estimator MD 4N is superior to all other estimators except T i(d)N for the neutro- sophic population, with a moderate or low correlation between the research variable and the supplementary variable (regardless of correlation is positive or negative).

Simulation study
We evaluate the suggested estimators' efficiency using simulated neutrosophic data, such as Y N and X N are neutrosophic random variates (NRV).We generate two sets of neutrosophic random numbers of N = 1000 , which are x ′ N and y ′ N from neutrosophic bivariate gamma distribution using the R programming language.Additionally, motivated by the simulated population generation strategies used by 55 , we generate the  11 summarises the findings of the simulated data set utilized to evaluate the suggested estimators' efficiency to that of traditional estimators under neutrosophic statistics.Tables 12, 13, 14, 15 and 16 contain the percent relative efficiency of neutrosophic estimators.The analysis by simulated data also verifies that T i(d)N , is the most efficient estimator.The simulation results suggest that the neutrosophic generalized estimator T i(d)N , produces more accurate and precise findings than other estimators.All estimators are unbiased (up to the first order of approximation), efficient, and reliable.
In Figs. 4 and 5, we have displayed the performance of simulated data by using boxplot and Q-Q plot, respectively.The boxplots show that data is positively skewed, which implies that the median is closer to the lower or bottom quartile.The boxplot with points beyond the whiskers indicates that the data has a few outliers.Normal Q-Q plot displays that the top end of the Q-Q plot deviates from the straight line, while the lower end follows the straight line, and that the curves have a more prominent tail to the right, indicating that they are right-skewed (or positively skewed).

Conclusion
Our study introduced neutrosophic estimators for accurately estimating the population median in datasets with uncertain, unclear values.Through precise additional variable measurements under simple random sampling, we developed improved neutrosophic estimators, evaluated them for bias and MSE, and demonstrated their superiority.Our proposed estimators offer the advantage of modelling uncertainty and vagueness inherent in many real-world scenarios, allowing for more flexible and nuanced decision-making processes.However, their disadvantage lies in the complexity of mathematical models and computational processes required, which can lead to increased computational costs and challenges in interpretation and precise quantification.We recommend these advanced estimators for future applications and highlight the ongoing need for research to enhance estimator effectiveness for various neutrosophic data types and sampling methods.Furthermore, future work will extend to multiple sampling designs, such as systematic, successive, and double sampling.

Figure 1 .
Figure 1.Workflow of the parameter estimation.

Figure 3
Figure3elaborates on the pivotal role of Q-Q (quantile-quantile) plots in statistics.These plots facilitate the graphical comparison of two probability distributions through their quantiles.They are instrumental in determining the distribution type of a random variable, spotting outliers, and assessing skewness.By plotting theoretical quantiles against sample quantiles, Q-Q plots reveal distribution traits, including skewness.Notably, deviations in the plot's upper end and a pronounced right tail indicate a right-skewed distribution, as demonstrated in standard Q-Q plot interpretations.In addition, the following formulae are used to get the percentage relative efficiency (PRE)

Figure 2 .
Figure 2. Boxplot of Populations I and II for each variable.

Figure 3 .
Figure 3. Normal Q-Q plot of Populations I and II for variable X N and Y N .

Figure 4 .
Figure 4. Boxplot of simulated data for each variable.

Figure 5 .
Figure 5. Normal Q-Q plot of simulated data for variable X N and Y N .

Table 1 .
Some real-life problems and domains under neutrosophic logic.
Evaluating environmental impacts with imprecise data, aiding in more effective decision-making Social sciences Opinion mining and sentiment analysis Analyzing sentiments and opinions in social media data, where opinions can be indeterminate or inconsistent Robotics Robot localization and navigation Improving robot navigation in uncertain environments by dealing with imprecise sensor data Economics Economic forecasting Enhancing economic forecasting models by incorporating uncertainty in economic data www.nature.com/scientificreports/

Table 3 .
Descriptive statistics of populations for single auxiliary variable.

Table 5 .
PRE's of proposed neutrosophic estimators T ⊖ i(d)N to M0N .

Table 6 .
PRE's of proposed neutrosophic estimators T ⊕ i(d)N to M0N .

Table 7 .
PRE's of proposed neutrosophic estimators T ⊗ i(d)N to M0N .

Table 8 .
PRE's of proposed neutrosophic estimators T ⊛ i(d)N to M0N .

Table 9 .
PRE's of proposed neutrosophic estimators T ⊚ i(d)N to M0N .

Table 10 .
Descriptive statistics for simulation study.

Table 13 .
PRE's of proposed neutrosophic estimators T ⊕ i(d)N to M0N .

Table 14 .
PRE's of proposed neutrosophic estimators T ⊕ i(d)N to M0N .

Table 15 .
PRE's of proposed neutrosophic estimators T ⊕ i(d)N to M0N .

Table 16 .
PRE's of proposed neutrosophic estimators T ⊕ i(d)N to M0N .